A Time-Based Solution for the Graph Isomorphism Problem

Abstract

This paper is devoted to the problem of isomorphism in graphs and proposes a method based on time response of a differential equation. First it is shown that the solution of a differential equation obtaining from a Laplacian matrix can be used as an index and the proof is presented. Then a search algorithm is proposed to find out that the two graphs are isomorphic and there is a permutation matrix describing relations between the graphs. The search algorithm depends on eigenvalues of the Laplacian matrix. For a Laplacian matrix with repeated eigenvalues, Greshgorin theorem is used to convert it to a matrix with non-repeated eigenvalues. This is performed by adding loops to vertices, so that they have separated Greshgorin bands. Then the time response of the differential equation is checked. The proposed method is performed on co-spectral graphs and results are described.